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Unformatted text preview: 88 LECTURE 8. THE MODULAR CURVE 8.6 Finally we interpret the spaces M k () as the spaces L ( D ) for some D on the Riemann surface X (). To state it in a convenient form let us generalize divisors to admit rational coefficients. We define a Q-divisor as a function D : X Q with a finite support. We continue to write D as a formal linear combination D = P a x x of points x X with rational coefficients a x . The set of Q-divisors form an abelian group which we shall denote by Div( X ) Q . For any x Q we denote by b x c the largest integer less or equal than x . For any Q-divisor D = P a x x we set b D c = X b a x c x. Theorem 8.7. Let D = 1 2 r 2 X i =1 x i + 2 3 r 3 X i = r 1 +1 x i + r X i =1 c i , D c = D- 1 k r X i =1 c i , where x 1 ,...,x r 1 are elliptic points of order 2, x r 1 +1 ,...,x r 1 2+ r 2 are elliptic points of order 3 and c 1 ,...,r are cusps. There is a canonical isomorphism of vector spaces M k () = L ( kK X + b kD c )) , M k ()...
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.

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